ATC + approaches.
There isn't a 'recording' apart from ATC tapes of what aircraft shot what approaches. For example if a plane shoots an approach, goes around and then diverts out, it'll likely be in the tower log, and potentially show as an approach in airport stats, but there isn't an admin log that says 'Blogs Air' flew two approaches to ILS 123, that is then subsequently forwarded to 'the authorities'.
CAP413 in DB6's post identifies that after a crew is identified as having descended past the 1,000ft for approach XYZ then reporting action (in the UK) will be taken. As mentioned previously ATC have to trust the crews to be truthful about their capability. An ILS radiates a signal capable of use by a Cat I/II/III equiped/qualified aircraft/crew. If a light twin says it's CatIII capable/qualified, it may seem unusual to a controller but they are unable to question it. They may emphasise the low RVR values as the APP controller appears to have done in the prelim report, in the hope that may be an aide memoir to a busy crew. I'd have thought a Metro making a Cat III approach very unusual, but it wouldn't have the same 'I wonder' factor if they were making a Cat II approach, as that type could reasonably be Cat II capable, based on my lack of knowledge, and an assumption on that same basis.
Unless you know the config/specs of each type you can't be the policeman, and I doubt many would want to be. At a busy unit in LVP's a controller has far more to think about than is the crew/aircraft mix 'right', so they have to trust the operators to be truthfull, just as the crew expect the RVR's to be accurately passed and full ILS airfield protection to be in place in low vis. Likewise a 737 can be Cat II or III, some operators put the RVR minima on the plan, but that doesn't show on the controllers printed strip, and I doubt is shown on an TWR/APP EFD display either.